### 15&22 (Porcupine) notation

Posted:

**Tue Oct 11, 2016 2:28 pm**There are two main approaches to notating linear temperaments:

1. MOS nominals, where Latin letters other than A to G, or Greek letters, are used to represent the notes in a Moment of Symmetry scale having typically 4 to 10 notes. This requires a non-standard staff in many cases, and isn't discussed in this post.

2. Chain-of-fifths (CoF) nominals, which uses only the nominals A to G, as a chain of fifths F C G D A E B, or equivalently as a chain of fourths B E A D G C F. This uses a standard staff.

See https://www.prismnet.com/~hmiller/music ... upine.html.

Porcupine has a generator of 160 to 165 cents (called a quill). Porcupine is commonly embedded in 15-edo, 22-edo or (their sum) 37-edo, so we can't assume it will be embedded in any particular EDO. In fact we can't assume it will be a closed tuning at all. The first thing we need to know, to generate a CoF-nominals notation for any linear temperament, is how many generators there are to its approximation of a 2:3 fifth, or a 3:4 fourth. Three quills make a fourth. This tells us that a chain of quills will be notated as 3 interleaved chains of fourths (as you can see below).

Then we need an accidental symbol to indicate which of the three chains of fourths a note is on. We'd like that symbol to represent a comma for a low prime number, or a simple ratio if possible. It turns out that the 5-comma 81/80 will do the job, because it contains 3 to the power 4, and 5 to the power -1, and Porcupine has -3 generators to the prime 3, and -5 generators to the prime 5, and -3 x 4 + -5 x -1 = -7, and you can see below, that adding to a note corresponds to jumping 7 places backwards on the chain of generators.

The following shows vertically, 37 notes in a chain of Porcupine generators (quills) using chain-of-fifth nominals. I've offset some of them so you can see the 3 interleaved chains of fourths. The Porcupine[7] MOS scale with the simplest notation is shown with its nominals in red.

1. MOS nominals, where Latin letters other than A to G, or Greek letters, are used to represent the notes in a Moment of Symmetry scale having typically 4 to 10 notes. This requires a non-standard staff in many cases, and isn't discussed in this post.

2. Chain-of-fifths (CoF) nominals, which uses only the nominals A to G, as a chain of fifths F C G D A E B, or equivalently as a chain of fourths B E A D G C F. This uses a standard staff.

See https://www.prismnet.com/~hmiller/music ... upine.html.

Porcupine has a generator of 160 to 165 cents (called a quill). Porcupine is commonly embedded in 15-edo, 22-edo or (their sum) 37-edo, so we can't assume it will be embedded in any particular EDO. In fact we can't assume it will be a closed tuning at all. The first thing we need to know, to generate a CoF-nominals notation for any linear temperament, is how many generators there are to its approximation of a 2:3 fifth, or a 3:4 fourth. Three quills make a fourth. This tells us that a chain of quills will be notated as 3 interleaved chains of fourths (as you can see below).

Then we need an accidental symbol to indicate which of the three chains of fourths a note is on. We'd like that symbol to represent a comma for a low prime number, or a simple ratio if possible. It turns out that the 5-comma 81/80 will do the job, because it contains 3 to the power 4, and 5 to the power -1, and Porcupine has -3 generators to the prime 3, and -5 generators to the prime 5, and -3 x 4 + -5 x -1 = -7, and you can see below, that adding to a note corresponds to jumping 7 places backwards on the chain of generators.

The following shows vertically, 37 notes in a chain of Porcupine generators (quills) using chain-of-fifth nominals. I've offset some of them so you can see the 3 interleaved chains of fourths. The Porcupine[7] MOS scale with the simplest notation is shown with its nominals in red.

G A B C D E F G A B C D E F G A B C D E F G A B C D E F G A B C D E F G A